The following notes are intended for our students following A-Level Modular
Physics, Cosmology, UCLES. They are intended as notes to clarify certain aspects
of the course. They are not intended to be a replacement for their own notes,or
to take the place of standard textbooks.

HTML is not a nice medium for using Greek symbols, so I have
substituted some of the standard symbols with appropriate English letters.

3. The Standard Model of the Universe

3.1 Line spectrum

When gases are heated they absorb energy. Electrons move to higher
energy levels (greater potential energy) Electromagnetic radiation is emitted
when an electron returns to a lower energy level. The energy emitted is equal
to the difference in energy levels between the two states. This is the photon
energy given by E = hf , and the corresponding wavelength is given by
w = c/f .

When light from a heated gas is analysed using a
spectrometer the different wavelengths are clearly visible as a set of
distinct lines. This is the emission spectrum of that gas. Each gas is
characterised by a unique set distinct lines (wavelengths) . This is a result
of the differences in the atomic structure of each element. No single
wavelength can identify an element , but the overall pattern clearly
identifies the element (or compound) This is of immense importance in studying
spectra from distant star.

All stars emit a continuous spectrum of all
wavelengths through the electromagnetic spectrum. The relative intensity of
these wavelengths is given by the black body distribution. Gases in the
surrounding 'atmosphere' absorb amounts of electromagnetic energy which
correspond to the differences between energy levels for the electrons . As
electrons return to the lower state the wavelengths are emitted in all
directions. Therefore very little of the original electromagnetic energy at
these wavelengths actually reaches the observer. The continuous spectrum of a
star appears to be missing these wavelengths . This pattern of missing
wavelengths in a continuous spectrum is called an absorption spectrum.
For a given gas the position of these absorption lines coincides exactly with
the wavelengths of the emission spectrum for that gas. Therefore , the study
of the spectra from distant stars can be used to identify gases surrounding
the star, by looking for characteristic 'fingerprint' pattern of gases and
elements

The absorption spectrum of a distant galaxy can be used to
determine the velocity v at which the galaxy is moving relative to our galaxy.
This is because each distinct pattern of wavelengths will have been 'red
shifted'...a consequence of the Doppler effect.The degree of red shift is
directly proportional to the recessional velocity.

3.2 The Doppler Effect

The change of wavelength due to the relative motion between the source and
observer is known as the Doppler Effect. We will make the assumption that
v is much smaller than c . Let w be the wavelength when there is no
relative velocity between source and observer, The change of wavelength is
directly proportional to the relative velocity v between the source and the
observer. The change of wavelength, = (v x w)/c and the
fractional change of wavelength = v/cAs v becomes larger
this approximation is no longer valid, due to the greater influence of
relativity.

3.3 Hubble's Law

One of the most important discoveries occured in 1929, when Edwin
Hubble discovered that the Universe is expanding. He found that the red-shifts
of a set of local galaxies were directly proportional to the distance
of these galaxies from our galaxy. This has became known as Hubble's
Law, and is given by

v = Hd

where H is known
as Hubble's constant. The value of H is measured in km/s per Megaparsec.
A value of 50 km/s per Mpc would mean that a galaxy 1Mpc away would be
receding at 50km/s. A galaxy 10 Mpc away would be receding at 500 km/s.

This equation does not take account of the effect of gravity on the
outward expansion. The presence of gravity will oppose the outward
expansion,so the velocity of expansion reduces with time. The value of H is
the value of H as measured today. It is sometimes called the Hubble
Parameter...since its value is not constant with time. ( A graph of
distance between two galaxies against time is not a straight line)

The value of H will always give a calculated value of the age of the
Universe (1/H) which is higher than the actual value. An accurate
determination of H depends on the accuracy to which we can determine distances
to distant galaxies.Unfortunately, the greater the distance of a galaxy the
greater the uncertainty in measurements of that distance.Most estimates give
values between 40 and 80 km/s per Megapasec.

The Fate of the Universe

The gravitational attraction between masses slows down the outward
expansion of the Universe. If we knew the mass of the Universe with a high
degree of accuracy , we would be able to predict confidently just what the
future of the Universe will be. Unfortunately much of the mass of the Universe
appears missing...or at least it hasn't been found yet. This missing mass,
which would enable the density of the Universe to be calculated , is difficult
to observe, unlike stars which emit their own light. Yet an accurate
determination is absolutely essential if we are to predict with some accuracy
the fate of the Universe.

3.5 The Cosmological Principle

According to the Cosmological Principle there are no preferred places
in the Universe. Measurements of the Universe made from Earth, disregarding
local irregularities, can be considered to be identical to those made in any
other part of the Universe. A classic illustration of this is in the way in
which the expansion of the Universe, according to Hubble's Law, occurs.

Take a length of elastic.Mark dots at even intervals to indicate
galaxies. Gradually pull the elastic apart.At any time the distances from a
galaxy will vary according with Hubble's Law, irrespective of which galaxy
(dot) is chosen as the reference point. A similar pattern can be observed by
using a series of dots on a balloon which is gradually inflated.

Therefore, irrespective of where we are in the Universe, we should
observe the same expansion properties , and the same laws of physics in
operation. Earth has therefore no special significance, other than to humans,
on a cosmic scale !

The Universe before t=0.01s : Some Problems to Solve !

The early Universe,according to the Big Bang, involved extremely high
temperatures. So high are these temperatures, that it is extremely difficult
to reach them in scientific experiments. Fundamental investigations into
particles involve accelerating particles at very high speeds, which would
normally be associated with particles at very high temperatures. It is
possible to reach speeds in particle accelerators which correspond to 10^15 K
, which would occur about 10^-15 s after the Big Bang. For times earlier than
this, we need to rely on theoretical models,used to explain our present
observations ,and 'extrapolate' these backwards to predict the mechanisms
which occur in the very early stages of the Big-Bang. This is an incredible
difficult task, because a number of complex interactions which take place .
Protons and neutrons are thought to consist of smaller particles, called
quarks At temperatures above 10^7 these quarks cannot be held together as
protons and neutrons. At higher temperature the four fundamental forces become
almost indistinguishable from one another . Different 'messenger' particles
are believed to be responsible for these four different forces. Above 10^5 K
photons, massless particles, are being converted into particles,and their
anti-particles.The reverse process is also happening . The theories are
therefore very different to theories involving everday experiences of
forces,and the synthesis of the four fundamental forces makes them
extraordinarily complex theories. These are known as Grand Unified Theories
(GUTs). These attempt to explain the physical behaviour of particles and
forces during the very early stages of the Universe, in a single set of
equations.The more recent theories which try to reach the goal of TOE (Theory
of Everything) are called String Theories. These have gained some success in
providing a desription of gravity, but the theory itself appears to some
scientists as a mathematical instrument or 'toy', with very little foundation
based on physics (as most of us know it !)

At time t=0 a
singularity is predicted, where all laws as we know them break
down.Space and time are infinitely distorted. The uncertainty principle
of quantum mechanics prevents any accurate predictions to be made for times
less than 10^-45 s.

Microwave Background Radiation

In 1965 , Arno Penzias and Robert Wilson discovered , quite accidently, the
existence of a low level microwave radiation. This was found to be virtually
of the same intensity no matter which way they looked with their radio
telescope. This was one of the greatest discoveries for the support of the Big
Bang Theory. After billions of years the great fireball would have cooled to
give a black-body peak temperature of approximately 2.7 K. Furthermore,
this temperature would be associated with a peak wavelength of 1.07mm .
Analysis of the radiation supported this prediction. Until recently, the
uniformity of the background radiation was of some concern to many
cosmologists.The apparent uniformity of the background radiation was not
consistent with the requirements for the formation of galaxies. Small
variations in the density are required in order that galaxies can be formed.
This is due to the effect of gravitational attraction between particles of
gases in these higher density regions. Therefore, this should be reflected in
slight variations in the background radiation. The measurement of such
variations was performed by Cosmic Background Explorer (COBE) in 1992.These
measurements amounted to variations of just 30 millionths of a degree Kelvin.